Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $p = \dfrac{3(q - 9)}{-7} \div \dfrac{q - 9}{q} $
Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{3(q - 9)}{-7} \times \dfrac{q}{q - 9} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 3(q - 9) \times q } { -7 \times (q - 9) } $ $ p = \dfrac {q \times 3(q - 9)} {-7 (q - 9)} $ $ p = \dfrac{3q(q - 9)}{-7(q - 9)} $ We can cancel the $q - 9$ so long as $q - 9 \neq 0$ Therefore $q \neq 9$ $p = \dfrac{3q \cancel{(q - 9})}{-7 \cancel{(q - 9)}} = -\dfrac{3q}{7} = -\dfrac{3q}{7} $